Mean field limits of a class of conservative systems with position-dependent transition rates
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917572712071168 |
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| author | Xue, Xiaofeng |
| author_facet | Xue, Xiaofeng |
| contents | In this paper, we are concerned with a class of conservative systems including asymmetric exclusion processes and zero-range processes as examples, where some particles are initially placed on $N$ positions. A particle jumps from a position to another at a rate depending on coordinates of these two positions and numbers of particles on these two positions. We show that the hydrodynamic limit of our model is driven by a nonlinear function-valued ordinary differential equation which is consistent with a mean field analysis. Furthermore, in the case where numbers of particles on all positions are bounded by $\mathcal{K}<+\infty$, we show that the fluctuation of our model is driven by a generalized Ornstein-Uhlenbeck process. A crucial step in proofs of our main results is to show that numbers of particles on different positions are approximately independent by utilizing a graphical method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mean field limits of a class of conservative systems with position-dependent transition rates Xue, Xiaofeng Probability In this paper, we are concerned with a class of conservative systems including asymmetric exclusion processes and zero-range processes as examples, where some particles are initially placed on $N$ positions. A particle jumps from a position to another at a rate depending on coordinates of these two positions and numbers of particles on these two positions. We show that the hydrodynamic limit of our model is driven by a nonlinear function-valued ordinary differential equation which is consistent with a mean field analysis. Furthermore, in the case where numbers of particles on all positions are bounded by $\mathcal{K}<+\infty$, we show that the fluctuation of our model is driven by a generalized Ornstein-Uhlenbeck process. A crucial step in proofs of our main results is to show that numbers of particles on different positions are approximately independent by utilizing a graphical method. |
| title | Mean field limits of a class of conservative systems with position-dependent transition rates |
| topic | Probability |
| url | https://arxiv.org/abs/2401.12512 |